Home
Class 12
MATHS
Find the image of point (1,2,3) in the l...

Find the image of point `(1,2,3)` in the line `(x-6)/3=(y-7)/2=(z-7)/(-2)dot`

Text Solution

Verified by Experts

Let the image of point `A(1, 2, 3)` in the line `l` whose equation is
`" " (x-6)/(3)=(y-7)/(2)=(z-7)/(-2)=k` (say) `" "` (i)
be `A'`. Then `AA'` is perpendicular to `l` and the point of intersection of `AA'` and `l` is the midpoint of `AA'`. Note that `M` is the foot of perpendicular from `A` to `l`.
The coordinates of any point on the given line are of the form `(3k +6, 2k+7, -2k+7)`. Therefore, the coordinates of `M` are `3k+6, 2k+7 and -2k+7` for some value of `k`. The direction ratios of `AM` are `3k+6-1, 2k+7-2 and -2k+7-3 or 3k+5, 2k+5, -2k+4`.
Also, the direction ratios of `l` are 3, 2 and -2.
since `AM bot l, a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)=0`.
`rArr" "3(3k+5)+2(2k+5)-2(-2k+4)=0`
or `" "17k+17=0 or k=-1`
Thus, the coordinates of `M` are 3, 5 and 9.
Suppose coordinates of `A'` are `x, y andz`,
The coordinates of the midpoint of `AA'` are
`(x+1)/(2), (y+2)/(2) and (z+3)/(2)` .
But the midpoint of `AA'` is (5, 3, 9). Therefore,
`" "(x+1)/(2)=3, (y+2)/(2) =5 and (z+3)/(2)=9`
`rArr" "x=5, y=8, z=15`
Thus, the image of `A` in `l` is (5, 8, 15).
Promotional Banner

Topper's Solved these Questions

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 3.3|19 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 3.4|5 Videos

  • THREE-DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 3.1|12 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE ENGLISH|Exercise All Questions|294 Videos

  • TRIGONOMETRIC EQUATIONS

    CENGAGE ENGLISH|Exercise Archives (Matrix Match Type)|1 Videos

Similar Questions

Explore conceptually related problems

Find the image of the point (1, 2, 3) in the line (x-6)/(3)=(y-7)/(2)=(z-7)/(-2) .

Find the image of the point (1, 2, 3) in the line (x-6)/(3)=(y-7)/(2)=(z-7)/(-2) .

Find the image of the point (1,6,3) in the line x/1=(y-1)/2=(z-2)/3

Find the image of the point (1,6,3) in the line x/1=(y-1)/2=(z-2)/3

Find the length of the perpendicular from the point (1,2,3) to the line (x-6)/3=(y-7)/2=(z-7)/(-2)dot

Find the image of the point (0,2,3) in the line (x+3)/(5)= (y-1)/(2) = (z+4)/(3) .

Find the image of the point (1,6,3) in the line x/1=(y-1)/2=(z-2)/3 . Also, write the equation of the line joining the given point and its image and find length of the segment joining the given point and its image.

Find the co-ordiantes of the foot of perpendicular drawn from the point (1,2,3) to the line (x-6)/(3) = (y-7)/(2) = (z-7)/(-2) .

Statement-I The point A(1, 0, 7) is the mirror image of the point B(1,6, 3) in the line (x)/(1)=(y-1)/(2)=(z-2)/(3) . Statement-II The line (x)/(1)=(y-1)/(2)=(z-2)/(3) bisect the line segment joining A(1, 0, 7) and B(1, 6, 3) .

The length of the perpendicular drawn from (1,2,3) to the line (x-6)/3=(y-7)/2=(z-7)/(-2) is a. 4 b. 5 c. 6 d. 7